padicValuation_eq_one_iff — Mathlib

English

For any prime p and integer x, v_p^{ℤ}(x) = 1 if and only if p ∤ x.

Русский

Для любого p и целого x, v_p^{ℤ}(x) = 1 тогда и только тогда, когда p не делит x.

LaTeX

$$v_p^{ℤ}(x) = 1 \\Leftrightarrow p \\nmid x$$

Lean4

theorem padicValuation_eq_one_iff {p : ℕ} [Fact p.Prime] {x : ℤ} : Int.padicValuation p x = 1 ↔ ¬(p : ℤ) ∣ x :=
  by
  simp only [← Rat.padicValuation_cast, Rat.padicValuation, Valuation.coe_mk, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk,
    Rat.intCast_eq_zero, padicValRat.of_int]
  split_ifs
  · simp_all
  · rw [← exp_zero, exp_injective.eq_iff]
    simp_all [Nat.Prime.ne_one Fact.out]

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